1、大学物理（热学）1)Mean Free Path2)Diffusion,Thermal conductivity,and Viscosity2006-10-12Equilibrium distributionNvvvvN222212+=dn is the number of moleculesxxxvvvd+vyvy+dvyzzzvvvd+in=NnvNnviiid22Nvnvn+=222211ni the number of molecules ofivnear vamong N molecules,how many are near vprobability that one molecu
2、le is near vproportional to dvx.()xxvvfdMeaning of dn/N()()()ddd),(dzzyyxxzyxvvfvvfvvfNvvvn=Isotropic assumption()()()()()2222zyxzyxvvvvvfvfvf+=()()22 exp xxvCvf=The solution isThe probability should be normalized=Nnd1()3223dexpd1=xxvvCNn=xxvvCdexp122()=xxvvCdexp12212=1each normalized()d d dexp1),(d
3、22223zyxzyxzyxvvvvvvNvvvn+=?=mTkv 3B2=Known 22223),(d=Nvvvnvvzyxwhile()()zyxzyxzyxvvvTkvvvmTkmNvvvnddd2exp2,dB22223B+=For unit volume(),dzyxvvvnMaxwell velocity distribution()zyxzyxvvvTkvvvmTkmnddd2exp2B22223B0+=vvTkmvTkmnvnd421exp2)(d2B223B0=Maxwell speed distributionFor one molecule,what speed is
4、most possible?Most probable speedvvvfnd4)(20Among N molecules,what is the speed most molecules have in unit interval?021expddB22=TkmvvvmTkvBm2=vvvvvNvndexp14)(d22m2232m=Maxwellian can be written asThe average speed()vvvvvvvd exp1422m20232m=mTk=B8xxvxde 4203m=mTkmTkvBBm41.12=mTkmTkvBB59.18=mTkmTkvBBr
5、ms73.13=xDJm =ntdN)(2=vIn an interval of t,number of collisions isThen mean free path iscylNcylL=21dn=ntdt r2=vvwhere the mean relative speed is21rvv=vcylNcylL=TkB8rvmTk=B82v2=ReferenceL.E.Reichl,A Mordern Course in Statistical Physics.University of Texas Press,Austin 1980 pp458optionaldifferent fro
6、m dimensionThen mean free path is221dn=Only a factor2analysis result.xTJQ =thermal conductivityFouriers lawConductionTransport phenomenaHeat transfer:conduction convection thermal radiationdiffusionviscosityDiffusionxDJm =xvJyp =Ficks law6ViscosityMicroscopic theory of transport coefficientsymvTkvmm
7、 ,2321 ,B2=pQM ,=)()()()(dd61d+=xxnxxntvAFor mass transport,=mn()()+=xmnxmntvAMdd61d=xtvA2dd61In general:=xvtAMJM31dddvD31=TCTknVB23 VDC=D=Similarly n21=r=1.121010msm1069.1 8m1068.2 kg1033.33B325B27=mTkvTkpnmHydrogen under STPm1067.17=2)2(rK)(mW1071.8,sm1042.9225=Dsm1028.1=25DExperimental results:)K
8、(mW1068.1=2大学物理（热学）Maxwell-Boltzmann Distribution2006-10-30MaxwellMaxwell-BoltzmannBoltzmann分布率的碰撞数等于反过来的逆过程的碰撞数时才能发生.现在他考虑的是两个粒子碰撞的速度，而不是一个粒子的不同速度分量，存在统计的独立性.1866年，Maxwell承认他1860年的推导“may appear precarious”.)()(),(2121vvvvffF=接着，他指出平衡态的达到，只有在初始态),(21vv),(21vv碰到末态这样他又给出了另一个推导.这就意味着),(),(2121vvvv=FF即)()()()(2121vvvv=ffff同时22221122221121212121vvvv+=+mmmm类似于前面的推导，他得到这时能量守恒的条件应该是1868年，Boltzmann将这种考虑推广到存在外力的情况，)exp()(22233vv=Nf其中mTkB22=2221211222121121)(2121)(21vvvv+=+mxVmmxVm即在势函数V(x)的情况.然后Boltzmann采用Maxwell的推导方式，得到+=)(21exp)(2xVhcfmvv其中TkhB1=这就是著名的Maxwell-Boltzmann分布率.
